- Open Access
On the absence of global solutions for some q-difference inequalities
© The Author(s) 2019
- Received: 20 October 2018
- Accepted: 22 January 2019
- Published: 30 January 2019
In this paper, we obtain sufficient conditions for the nonexistence of global solutions for some classes of q-difference inequalities. Our approach is based on the weak formulation of the problem, a particular choice of the test function, and some q-integral inequalities.
- Global solution
- q-difference inequalities
The study of sufficient conditions for the nonexistence of global solutions to differential equations or inequalities provides important information in theory as in applications. First, sufficient conditions for the absence of solutions provide necessary conditions for the existence of solutions. Second, useful information on limiting behaviors of many physical systems can be obtained via the nonexistence criteria. Indeed, having an information on the blowing-up of solutions can help in preventing accidents and malfunction in industry. It helps also in improving the performance of machines and extending their lifespan.
There are several works in the literature concerning the nonexistence of solutions for different classes of differential equations or inequalities involving nonstandard derivatives. In particular, the study of the absence of solutions for different types of fractional differential problems has received a great attention from many researchers. In this direction, we refer the reader to [15, 16, 18–21] and the references therein. However, to the best of our knowledge, there are no investigations on the nonexistence of solutions in quantum calculus.
The q-difference calculus or quantum calculus is an old subject, which is rich in history and in applications. It was initiated by Jackson [11, 12] and developed by many researchers (see, e.g., [1, 6, 8]). We can find in the literature several papers dealing with the existence of solutions for different kinds of q-difference equations; see, for example, [3–5, 9, 10, 13, 17, 24] and the references therein.
In this paper, we obtain sufficient criteria for the absence of global solutions to problems (1)–(2) and (3)–(4). The proofs are based on an extension of the test function method due to Mitidieri and Pohozaev  to quantum calculus.
The paper is organized as follows. In Sect. 2, we recall some basic concepts on q-calculus and present some properties and lemmas that will be used in the proofs of our results. Section 3 is devoted to study the nonexistence of global solutions for problem (1)–(2). In Sect. 4, we establish a nonexistence result for problem (3)–(4).
In this section, we recall some basic concepts on quantum calculus and provide some useful properties.
Next, we recall the following q-integration-by-parts rule.
B. Samet extends his appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia).
This work is supported by the Distinguished Scientist Fellowship Program (DSFP) at King Saud University.
The authors declare that they carried out all the work in this manuscript and read and approved the final manuscript.
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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